In terms of classical Bessel function, we represent explicitly the radial part of Whittaker functions on SU(2, 1) belonging to the large discrete series representations. Moreover we compute archimedean local L-factors corresponding to a construction of L-function by Gelbart and Piatetski-Shapiro.

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For some reaction-diffusion equations, Lyapunov-Schmidt reduction was shown to be applicable to construct singularly perturbed equilibrium solutions. For this application, it is indispensable to show that some inverse operator are uniformly bounded. In this paper, we give an elementary proof of this fact.

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We determine the character group of the infinite unitary group of a unital exact C^*-algebra in terms of K-theory and traces and obtain a description of the infinite unitary group modulo the closure of its commutator subgroup by the same means. The methods are then used to decide when the state space SK₀(A×_α\mathbb{Z}) of the K₀ group of a crossed product by \mathbb{Z} is homeomorphic to SK₀(A)_α. or T(A)_α. We also consider the crossed product A×_αG discrete countable abelian group G and give necessary and sufficient conditions for the equality T(A×_αG)=T(A)_α to hold.

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We determine the mod-p cohomology ring and some of the integral cohomology ring structure of a p-group expressible as an extension with kernel cyclic of order p and quotient C_p^m ⊗ C_pⁿ, where m, n>1.

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We study the high-energy behavior of the scattering amplitude and the total scattering cross section for a Dirac operator with a 4×4 matrix-valued potential. Moreover, in the electro-magnetic case, it is shown that the electric potential and the magnetic field can be reconstructed from the high-energy behavior of the scattering amplitude. The study of the high-energy behavior of the resolvent estimates is crucial for our proof.

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